I obviously can't answer your question about every culture, but I can discuss a few cultures that I personally know a bit about. I think it's a mistake to say that "every culture that used a sword" made it mythic. I'll provide a couple of counterexamples.

The first group I'd like to mention is the Greeks, in the Archaic and Classical periods. Greek hoplite soldiers serve as a fairly straightforward counterexample to the prominence of the sword in every culture. In hoplite warfare, swords were not particularly prominent; instead, the most symbolically-significant tools of war were the spear and the shield. A brief description of Greek warfare in this time is warranted to explain the significance of the spear and shield. Hoplites formed into a phalanx, a tightly-packed rectangular military formation, protected in the front by a shield wall. Each soldier in the front lines would hold a shield on his left arm, and a spear in his right.

The spear was, for this reason, the primary weapon: unless the two formations were right next to each other, soldiers would not be close enough to use a sword. Important military innovations centered not on the sword, but on the spear. For example, one of the main military innovations that allowed Philip of Macedon to conquer most of Greece was the adoption of the sarissa, a new spear much longer than ones used by other polis (13+ feet as opposed to ~7).

The shield held immense symbolic value. The shields of soldiers in the front lines overlapped; each soldier was protecting not only himself, but also his comrade to the right. Further, the shield carried the insignia of the hoplites' polis. For these reasons, the shield was perhaps the most important weapon or piece of armor symbolically. Losing one's shield in battle, for example, was seen as deeply shameful, as a sign of cowardice. This is the root of the quote from a Spartan mother, recorded in Plutarch's Sayings of Spartan Women, exhorting her son to come back "either with [his shield] or upon it."

A second group that to which the sword was not completely central was the samurai. This may sound strange; today samurai are remembered primarily as swordsmen, and it is true that from the mid 1500s onwards, swords were their main weapons. However, the main weapon of the early samurai was not the sword, but the bow and arrow. In the 1200s, for example, Takezaki Suenaga, commemorating his battles against the Mongols, described the samurai way as "the way of the bow and arrow." In this time, swords were used, but the bow and arrow was the primary weapon of the samurai in terms of symbolic significance.

Swords became tied to samurai identity only later. After the introduction of firearms into Japan in the 1500s, bows slowly fell out of use. The bow and arrow held some advantages over early firearms, but was much more difficult to use. Learning to shoot a bow accurately required a significant amount of training, while anyone could shoot a musket. Swords, however, remained useful in close quarters combat. For this reason, samurai identity became more and more attached to the use of the sword.

Samurai tied their identity to the bow and later the sword in large part because of the skill required to use these weapons. Samurai were valued as warriors; skill with the sword or the bow allowed a samurai to demonstrate his abilities to potential employers. In the 1600s, after the bow and arrow became mostly obsolete, samurai such as Miyamoto Musashi traveled around Japan challenging other samurai to sword duels, to establish martial superiority. In this way, skill with a sword became a samurai's main way to demonstrate his ability, and the sword became the main symbol of the samurai. Effective use of the sword distinguished a samurai not only from his peers, but from the lower-class as well. As mentioned above, almost anyone could use a musket, and cheap, lower-class mercenaries became more and more prominent in Japan after the introduction of gunpowder. The samurai's use of a sword allowed him to demonstrate his continued value, despite the emergence of new forms of military technology.

So I would not agree that swords were always, or even almost always the most "mythic" weapon in a culture. There are many other examples of societies that valued other weapons more than swords as well. It may just be a European thing, and perhaps the emphasis on the sword today comes from representations of medieval Europe in films and elsewhere.

u/alriclofgar has written about this here